reconfigurable optical computer for solving partial
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Reconfigurable Optical Computer for Solving Partial Differential Equation
by Chen Shen
B.S in Optoelectronic Science and Technology, July 2016, Shenzhen University
A Thesis submitted to
The Faculty of The School of Engineering and Applied Science
of The George Washington University in partial fulfillment of the requirements
for the degree of Master of Science
January 10, 2020
Thesis directed by
Volker J. Sorger Associate Professor of Engineering and Applied Science
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Acknowledgements
First, I would express my thanks to Professor Volker J. Sorger for teaching me with
extraordinary knowledge and experience, also giving me a wonderful research opportunity
as a master student. Also, I would like to thank all the members of the group. They shared
the new ideas, suggestions, and comments with me especially Dr. Mario Miscuglio, who
gives me opportunities to ask any scientific question anytime and tutors me with great
patience. I have asked him many questions and he has answered even more. Last, here, I
would also thank the staffs in The George Washington University (GW) Nanofabrication
and Imaging Center (GWNIC).
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Abstract of Thesis
Reconfigurable Optical Computer of Solving Partial Differential Equation
Nowadays, digital processors are overwhelmed by vast data and complex iterative
operation. Thus, some of the workloads can be taken by more specific processor whose
hardware is engineered to map specific functionalities and operations such as TPU and
FPGAs. In this category, various photonic engines1 which leverage on the absence of RC
delay, distributed network and possibility to do weighted addition inherently by
interference and concurrently exploit attojoule efficient high-speed modulators. Here we
introduce Reconfigurable Optical Computer (ROC), a research project undertaken by
Professor Volker J. Sorger’ OPEN Lab Team and professor Tarek El-Ghazawi’s HPCL
Team both at the George Washington University, through the funding of NSF RAISE
Grant. This project is mainly working on the physics theory and fabrication technology of
analog co-processors in silicon photonics and optical metatronics. The goal of ROC is to
investigate, model and ultimately develop an optical solution that can numerically and
experimentally solve various partial differential equations (PDE).
My contribution to ROC has been to prove the reconfigurability of the photonic
integrated version of ROC, by numerically modeling a programmable photonic chip in a
photonic interconnect simulation framework (Lumerical-Interconnect) photonic
integrated aiming to map different partial differential equation problems which describe
heat transfer problems in different materials. I compared the solutions obtained at each
node of photonic circuit to the discretized solutions obtained using commercially
available heat transfer simulation (COMSOL) which uses finite element modeling. Since
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each material has precise thermal property, the characteristic heat transfer map is
uniquely represented, thus attaining a library of photonic chip configuration for solving
Laplace equation applied at different domains. Therefore, the designed chip can be
reprogrammed to solve Laplace heat transfer equation for different materials and holds
the potential to solve other kinds of PDE.
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Table of Contents
Acknowledgements ........................................................................................................... ii
Abstract of Thesis ............................................................................................................ iii
Table of Contents ............................................................................................................. v
List of Figures .................................................................................................................. vi
List of Tables .................................................................................................................. vii
List of Abbreviation ...................................................................................................... viii
Glossary of Terms ............................................................................................................ x
Chapter 1 – Towards Optical Analog Computing ........................................................ 1
1.1 Gap of tradition electronic circuit ..................................................................... 1
1.2 Partial Differential Equation (PDE) .................................................................. 1
1.3 Reconfigurable Optical Computer (ROC) ........................................................ 3
1.4 Photonic Integrated Circuit (PIC) ..................................................................... 3
Chapter 2 – Reconfigurable part of ROC ...................................................................... 4
2.1 First year of ROC .............................................................................................. 4
2.2 Innovation of reconfigurability ......................................................................... 4
2.3 My contribution to ROC project ....................................................................... 5
2.2.3 Stage 1 Optical Mesh Building .................................................................. 5
2.3.2 Stage 2 Thermal simulation in COMSOL ................................................. 8
2.3.3 Stage 3 Normalization and Comparison .................................................. 12
2.3.4 Result ....................................................................................................... 13
2.3.5. Analysis................................................................................................... 16
Chapter 3 – Discussion .................................................................................................. 19
3.1 Limitation of ROC .......................................................................................... 19
Whole bibliography ........................................................................................................ 19
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List of Figures
Figure 1. (a)Image of node structure (b) General structure of optical simulation model. ..6
Figure 2. (a) A general structure of optical simulation (b)Image of node structure ............7
Figure 3. The geometric setting of thermal transfer simulation. .........................................8
Figure 4. Heat transfer simulation of Zirconium .................................................................9
Figure 5. The geometric setting of thermal transfer simulation. .......................................10
Figure 6. The heat map of Aluminum, temperature ranging from 0K to 104K. ................11
Figure 7. Models matched with Aluminum, Vanadium and Calbarb ...............................13
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List of Tables
Table 1. The excel file generated by python-scripted tool ............................................... 13
Table 2. The error sets are input into Origin Lab software .............................................. 16
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List of Abbreviation
EAM
electro-absorption modulator ................................................................................4
ER
extinction ratio .....................................................................................................5
FEM
finite element model ............................................................................................4
GW
The George Washington University .................................................................. iv
GWNIC
The George Washington University Nanofabrication and Imaging Center ....... iv
HPCL
High Perfomance Computing Laboratory ............................................................v
InP
Indium Phosphide ................................................................................................3
OPEN
Orthogonal Physics Enabled Nanophotonics .......................................................v
PDE
partial differential equation ..................................................................................v
ROC
Reconfigurable Optical Computer .......................................................................v
SPACE
Silicon Photonic Approximate Computing Engine ..............................................4
ix
S-parameter
scattering parameter .............................................................................................5
x
Glossary of Terms
Application-specific integrated circuit
An application-specific integrated circuit is an integrated circuit (IC) customized for
a particular use, rather than intended for general purpose use.
Co-processor
A co-processor is a computer processor used to supplement the functions of the
primary processor (the CPU). Operations performed by the coprocessor may be floating
point arithmetic, graphics, signal processing, string processing, cryptography or I/O
interfacing with peripheral devices. By offloading processor-intensive tasks from the
main processor, coprocessors can accelerate system performance.
COMPSOL Multiphysics
COMSOL Multiphysics is a cross-platform finite element analysis, solver and
multiphysics simulation software. It allows conventional physics-based user interfaces
and coupled systems of partial differential equations (PDEs). COMSOL provides an IDE
and unified workflow for electrical, mechanical, fluid, and chemical applications. An API
for Java and LiveLink for MATLAB may be used to control the software externally, and
the same API is also used via the Method Editor.
Distributed element model
In electrical engineering, the distributed element model or transmission line model
of electrical circuits assumes that the attributes of the circuit (resistance, capacitance, and
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inductance) are distributed continuously throughout the material of the circuit. This is in
contrast to the more common lumped element model, which assumes that these values are
lumped into electrical components that are joined by perfectly conducting wires. In the
distributed element model, each circuit element is infinitesimally small, and the wires
connecting elements are not assumed to be perfect conductors; that is, they have
impedance. Unlike the lumped element model, it assumes non-uniform current along
each branch and non-uniform voltage along each node. The distributed model is used at
high frequencies where the wavelength becomes comparable to the physical dimensions
of the circuit, making the lumped model inaccurate.
Epsilon-near-zero
The window of frequency in which the permittivity is low, i.e., near the plasma
frequency, has also become a topic of research interest in several potential applications.
The first attempts to build a material with low permittivity at microwave frequencies date
back to several decades ago, where their use was proposed in antenna applications for
enhancing the radiation directivity. Similar attempts have been presented over the past
years with analogous purposes. Other more recent investigations of the properties of ɛ-
near-zero (ENZ) materials and metamaterials and their intriguing wave interaction
properties have been reported. In particular, Ref. 18 addresses and studies the possibility
of designing a bulk material impedance matched with free space but whose permittivity
and permeability are simultaneously very close to zero. In all these works, the main
attempt has been to exploit the low-wave-number (index near zero) propagation in such
materials, which might provide a relatively small phase variation over a physically long
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distance in these media. When interfaced with materials with larger wave number, this
implies the presence of a region of space with almost uniform phase distribution,
providing the possibility for directive radiation toward the broad side to a planar
interface, as proposed over past the years for antenna applications. Also, in Ref. 18, the
possibility of utilizing the matched low- index metamaterial for transforming curved
phase fronts into planar ones has been suggested, exploiting the matching between the
aforementioned metamaterial and free space. Nader Engheta’s3 group has proposed
several different potential applications of ENZ and/or µ-near-zero materials for different
purposes. Relying on the directivity enhancement that such materials may provide.
Extinction ratio
The ratio re = P1/P0 of two optical power levels of a digital signal generated by an
optical source expressed as a fraction in dB or as percentage.
Finite difference method
In mathematics, finite-difference methods (FDM) are numerical methods for
solving differential equations by approximating them with difference equations, in which
finite differences approximate the derivatives. FDMs are thus discretization methods.
FDMs convert a linear (non-linear) ODE/PDE into a system of linear (non-linear)
equations, which can then be solved by matrix algebra techniques. The reduction of the
differential equation to a system of algebraic equations makes the problem of finding the
solution to a given ODE ideally suited to modern computers, hence the widespread use of
FDMs in modern numerical analysis1. Today, FDMs are the dominant approach to
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numerical solutions of partial differential equations.
Indium tin oxide
Indium tin oxide (ITO) is a ternary composition of indium, tin and oxygen in
varying proportions. Depending on the oxygen content, it can either be described as a
ceramic or alloy. Indium tin oxide is typically encountered as an oxygen-saturated
composition with a formulation of 74% In, 18% O2, and 8% Sn by weight. Oxygen-
saturated compositions are so typical, that unsaturated compositions are termed oxygen-
deficient ITO. It is transparent and colorless in thin layers, while in bulk form it is
yellowish to grey. In the infrared region of the spectrum it acts as a metal-like mirror.
Indium tin oxide is one of the most widely used transparent conducting oxides because of
its two main properties: its electrical conductivity and optical transparency, as well as the
ease with which it can be deposited as a thin film. As with all transparent conducting
films, a compromise must be made between conductivity and transparency, since
increasing the thickness and increasing the concentration of charge carriers increases the
material’s conductivity and decreases its transparency. Thin films of indium tin oxide are
most commonly deposited on surfaces by physical vapor deposition. Often used is
electron beam evaporation, or a range of sputter deposition techniques.
Lumerical INTERCONNECT
INTERCONNECT, Lumerical’s photonic integrated circuit simulator, verifies
multimode, bidirectional, and multi-channel PICs. Creating project in this hierarchical
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schematic editor, people can use our extensive library of primitive elements, as well as
foundry-specific PDK elements, to perform analysis in the time or frequency domain.
Lumped element
The lumped element model (also called lumped parameter model, or lumped
component model) simplifies the description of the behavior of spatially distributed
physical systems into a topology consisting of discrete entities that approximate the
behavior of the distributed system under certain assumptions. It is useful in electrical
systems (including electronics), mechanical multibody systems, heat transfer, acoustics,
etc. Mathematically speaking, the simplification reduces the state space of the system to a
finite dimension, and the partial differential equations (PDEs) of the continuous (infinite-
dimensional) time and space model of the physical system into ordinary differential
equations (ODEs) with a finite number of parameters.
Multigrid method
In mathematics, the conjugate gradient method is an algorithm for the numerical
solution of particular systems of linear equations, namely those whose matrix is
symmetric and positive-definite. The conjugate gradient method is often implemented as
an iterative algorithm, applicable to sparse systems that are too large to be handled by a
direct implementation or other direct methods such as the Cholesky decomposition. Large
sparse systems often arise when numerically solving partial differential equations or
optimization problems. The conjugate gradient method can also be used to solve
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unconstrained optimization problems such as energy minimization. It was mainly
developed by Magnus Hestenes and Eduard Stiefel who programmed it on the Z4. The
biconjugate gradient method provides a generalization to non-symmetric matrices.
Various nonlinear conjugate gradient methods seek minima of nonlinear equations.
Nanophotonics
Nanophotonics or nano-optics is the study of the behavior of light on the nanometer
scale, and of the interaction of nanometer-scale objects with light. It is a branch of optics,
optical engineering, electrical engineering, and nanotechnology. It often (but not
exclusively) involves metallic components, which can transport and focus light via
surface plasmon polaritons. The term nano-optics, just like the term optics, usually refers
to situations involving ultraviolet, visible, and near-infrared light (free-space wavelengths
from 300 to 1200 nanometers).
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Chapter 1 – Towards Optical Analog Computing
1.1 Gap of tradition electronic circuit
Although analog electronic computing engine performs extraordinarily in digital
systems for special-purpose processing because of its non-iterative operation nature, it is
limited by run-time and programmability-speed1. Digital technology is facing the
inevitable halt of Moore’s law due to physical limits which constrain denser integration
and the constraints given by Dennard scaling4 on power consumption which states that as
transistors get smaller, their power density stays constant, so that the power consumption
stays in proportion with area. The current improvement strategy is parallelization, using
multiple processors or co-processors with specific architectures which are able to
efficiently perform complex or recursive operations. All these technologies still rely on
electronic therefore limited by RC delay and temporization set by a <4 GHz clock.
We are using instead analog processor for solving a specific functionality which
usually requires immense power consumption and lengthy when implemented on analog
process. To accomplish higher processing data throughput/runtime ratio, here we
demonstrate photonic integrated circuit, which enables both analog compute-hardware
while exploiting time parallelism as well as capitalizing on-chip dense integration.
1.2 Partial Differential Equation (PDE)
In mathematics, a partial differential equation (PDE) that contains unknown
multivariable functions and their partial derivatives. PDEs are used to formulate problems
involving functions of several variables, and are either solved numerically, or used to
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create a computer model. PDEs can be used to describe a wide variety of phenomena
such as sound, heat, diffusion, electrostatics, electrodynamics, fluid dynamics, elasticity,
gravitation and quantum mechanics8. These seemingly distinct physical phenomena can
be formalized similarly in terms of partial differential equations. Just as ordinary
differential equations often model one-dimensional dynamical systems, PDEs often
model multidimensional systems and multivariable functions. PDEs find their
generalization in stochastic partial differential equations15.
For the Reconfigurable Optical Computer project, we are aiming to solve two-
dimensional second-order elliptic PDEs. Regarding elliptic PDEs, setting boundary
condition is significant for deriving PDEs because you can derive PDEs by applying
boundary condition to the originally equations. For example, in a 2-D Laplace’s equation:
𝛻𝛻2𝑢𝑢 = 𝜕𝜕2𝑢𝑢
𝜕𝜕𝜕𝜕2
𝜕𝜕2𝑢𝑢 + 𝜕𝜕𝜕𝜕2 = 0
for the geometry of a rectangle, 0 ≤ x ≤ L and 0 ≤ y ≤ H, the boundary condition will
be:
u(0, y) = g1(y) u(L, y) = g2(y) u(x, 0) = f1(x) u(x, H) = f2(x)
Then by adding these conditions to Laplace’s equation, we can solve it to:
u(x, y) = u1(x, y) + u2(x, y) + u3(x, y) + u4(x, y)
Because the Laplace’s equation is linear and homogeneous and each of the pieces is
a solution to Laplace’s equation then the sum will also be the solution. In my project, the
heat transfer equation is also a second-order partial differential equation, thus we can
build a analog computing engine mapping with heat transfer equation, indicating that we
can solve PDE in analog computer.
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1.3 Reconfigurable Optical Computer (ROC)
Reconfigurable Optical Computer (ROC) is a reconfigurable photonic accelerator,
implemented in two distinct physical technologies, capable of solving PDEs through the
finite difference method, constructed in a physical analog mesh which is distributed
network but differing from the electrical implementation which is usually lumped circuit
by performing summation of electrical displacement current for Metatronic ROC and
summation of optical intensity in photonic ROC9.
1.4 Photonic Integrated Circuit (PIC)
A photonic integrated circuit or integrated optical circuit miniaturizes multiple
photonic functions and is similar to an electronic integrated circuit. 57
The most commercially utilized material platform for photonic integrated circuit is
Silicon and Indium Phosphide (InP), which allows for the integration of various optically
active and passive functions on the same chip.
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Chapter 2 – Reconfigurable part of ROC 2.1 First year of ROC
Our group1 have recently established a Silicon Photonic Approximate Computing
Engine (SPACE) used for solving 2D second-order elliptic PDEs. Silicon Photonic
Invertible Cloverleaf Cross mesh (SPICC) is the physical implementation of a photonic
mesh able to drive the optical power similarly to a lumped electrical circuit which mimics
a Heat-transfer problem. The four-directional nodes in the model split the incoming light
equally into three other directions simulating an optical version of Kirchhoff’s Law in
analogy to a uniform electronic resistive circuit. By manipulating the intensity and
position of special light input, it is possible to set the boundary condition. The output at
each node, was spit adding an extra drop line and the optical power was coupled out by
means of grating couplers and ultimately an IR camera was used for detecting in parallel
the optical power at all the output ports which represents the solution of the discretized
solution given by the photonic circuit at each pint of the mesh. Furthermore, comparing
with a simulated heat transfer problem using finite element model (FEM) tool which
meshes the domain with the same resolution, they achieved a 97 percent accuracy PDE
solution.
2.2 Innovation of reconfigurability
According to the previous work mentioned above, the approximate optical
computing engine our group build is able to find accurate solutions to a PDE which is
fixed for the reason that the engine fabricated is passive. Although, it is envisioned that
the reconfigurability functionality can be obtained by adding electro-absorption
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modulator (EAM) at each node. My goal is to proceed in this direction enabling the
reprogrammable functionality to the optical computing engine. Before trying to solve
different type of PDEs, our system needs to be able to map the same heat transfer
equation for different domains characterized by various thermal conductivity and heat
capacitance.
Besides, in our photonic computing engine, boundary conditions can also be
reprogrammed for the accessibility of shining light at any node in the grid.
2.3 My contribution to ROC project
2.2.3 Stage 1 Optical Mesh Building
My initial work of ROC is building a new designed model, showing the
reconfigurability of the computing engine. I choose Lumerical Interconnect software to
emulate the PIC. As a first step we introduce reconfigurability, aiming to make the engine
tunable, by adding modulators between neighboring nodes. Within the numerical
simulation environment, the dynamic range of a modulator is translated in terms of
scattering parameter matrix (S-parameter, transmittance and reflectance). In this way, we
can simply tune the extinction ratio (ER) of modulator by directly altering the scattering
matrix of it. For instance, if a -10dB ER modulator would be introduced, first the -10dB
has to be converted to percentage, which is 10%. Since the S-parameter matrix describing
an N-port (in our case, 2-port) network that will be square matrix of dimension 2 and will
therefore contain 4 (22) elements, we need to take a square root of this 10% for S-
parameter. Thus, we can input the 0.316228 as the value of s21 amplitude for S-
parameter matrix in the Interconnect.
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In optical simulation, a 3x3 optical mesh was built to imitate the mesh structure of
heat transfer. At crossing nodes, we select another type of S-parameter to perform the
optical characteristic of equal splitter. Considering every node has four directions,
regardless which direction does the light propagate through this S-parameter, the light
will split into other three directions with equivalent value. To implement the function of
equal splitter, the matrix is read from a perfect equal splitter node file. In this file, the
script was written to set the incoming light evenly into three remaining directions. We
use power meter to read and record the power dropped at each node. A 1W continuous
wave laser source is placed at the left top corner to generate light which mimics the BC
of high local temperature. (Figure 1)
(a) (b)
Figure 1. (a)A detailed image of node structure. We can read and add the data of four power meters connected with an equal splitter around the node up to obtain the power through this node. (b) A general structure of optical simulation model, which is a 5x5 mesh. Light propagates from a continuous wave laser source located at left top corner.
The time window of whole model is set to be 500 picoseconds while the bit rate is
2.5bits/s. Also sample rate is fixed to 193.1 THz, which is also the frequency of every
waveguide. Hence, number of samples is 96550, calculated by other settings above.
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However, after simulation, several problems are revealed. First, the mesh size is so
micro that we could not observe the heat decay slope. However, considering the
simulation time boosts exponentially when mesh size increases, we can only rise the
mesh size to 5x5. On the other hand, the value recorded by power meter is unstable.
Therefore, we replaced it with oscilloscope, which would record the all 9998 values
during the simulating process.
Here we present a new model (Figure 2), which is 5x5 size and installing
oscilloscope instead of power meter.
Figure 2. (a) A general structure of optical simulation model, which is a 5x5 mesh. Light propagates
from a continuous wave laser source located at left top corner. (b)A detailed image of node structure. We can read and add the data of four oscilloscopes connected with an equal splitter around the node to obtain the power through this node.
Although we place modulators between every two neighboring nodes, the default
extinction ratio is set to be 1, which will not have influence on light transmission because
Pin/Pout ratio is also 1. These modulators will only be activated until we alter the
scattering matrix parameter.
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2.3.2 Stage 2 Thermal simulation in COMSOL
Next, a baseline with heat transfer simulation should be established. In COMSOL
Multiphysics simulation, five materials are selected to build the thermal baselines:
Aluminum, Copper, Gold, Silver, Zirconium. To study the heat map of these materials,
several configurations need to be set in advance. First, geometry (figure 3): In a two-
dimensional plane, one 100x100 square (default geometrical unit in COMSOL) is equally
divided into 25 blocks. Meanwhile, two point are placed to form an angle at left top
corner, these two sides of the angle are settled to be -173oC (100K). Simultaneously all
boundaries of external square are set to be minus 273oC (0K), which is the heat sink in
this model.
Figure 3. The geometric setting of thermal transfer simulation, a 5 by 5 square with one heat source
(-173oC) at the left top corner and heat sinks (-273oC) at remaining boundaries.
After simulation, we can have a heat map like the example of the heat transfer in a
Zirconium plate. To compare with optical simulation result, we need to extract 25 values
from the center of 25 blocks.
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Figure 4. The heat transfer simulation of Zirconium, with heat source at left top corner while heat dissipates at boundary heat sink.
Three problems are exposed in this simulation:
1. Since the heat source is on the edge of the square, that the highest value of
temperature would so be on the edge too. However, we extract thermal value from the
center of smaller blocks to read and draw the heat map. Here deflection is cause because
the highest value is not recorded. To solve this problem, we need to place the heat source
at the center of left top block that we can accurately present the heat map of heat transfer
in materials.
2. Regarding materials, because all five materials we choose are metal, the thermal
properties are highly analogous. Thus, we can hardly build different PICs mapping with
these metal’s thermal simulation. For preventing similarity, we need to focus on materials
with very different thermal conductive properties. Thus, we can have an easier
comparison with the photonic engine solution.
3. The heat source temperature versus temperature of heat sink is controversial, for
the reason that in COMSOL Multiphysics, thermal conductivity of various materials is
calculated as a function rather than a numerical constant. According to Vosteen and
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Hans-Dieter62, the general equation for the temperature dependence of thermal
conductivity can be formulated:
λ(T) = λ(0)0.99 + T(a − b/λ(0))
where λ is the thermal conductivity, a is the empirical constants and b is the
corresponding uncertainties. By deriving the equation, we can find out that if T=0K, this
equation does not hold.
In order to solve problems above, an advanced heat transfer model is made. First,
three materials are selected: Aluminum, Vanadium and Calcarb CVD 20 (a new insulator
material made up of carbon fiber). Geometrically, in a two-dimensional plane, one
100x100 square (default geometrical unit in COMSOL) is equally divided into 25 blocks
(Figure 5). Meanwhile, a 0.1x0.1 square with its center located at the center of top left
block is set be a 9727oC (10000K) heat source, simultaneously all boundaries of external
square are set to be 0oC, which are the heat sink in this model.
Figure 5. The geometric setting of thermal transfer simulation, which is a 5 by 5 square with one heat
source (9727oC) at the left top block center and heat sinks (0oC) at boundaries.
The computing engine of thermal simulation is described by following equation:
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d𝑧𝑧ρC𝑝𝑝𝐮𝐮 ∙ ∇T + ∇ ∙ 𝐪𝐪 = d𝑧𝑧Q + q0 + d𝑧𝑧Q𝑡𝑡𝑡𝑡𝑡𝑡 (1)
𝐪𝐪 = −d𝑧𝑧𝑘𝑘∇T (2)
Where ρ represents the density of material(kg/m3), Q is the heat source, u is the
velocity, q is the heat flux and 𝑘𝑘 is the thermal conductivity(W/(m∙k)). In our stationary
case, dz is out-of-plane thickness, C𝑝𝑝 represents the specific heat capacity(J/(kg∙k)), and
Qted is the thermoelastic damping.
The final setting is regarding mesh size, we choose free triangular mesh type with
maximum mesh size of 1 (default geometric unit in COMSOL) and minimum mesh size
of 1/100. In order to achieve the heat map, the mesh size must be smaller than the
minimum element size of my model, which is the heat source square. However, the
simulation time would exponentially increase if the mesh size is smaller than my setting.
After computing study, the heat maps for each material are presented (figure 6). We
record the thermal values of 25 points at the center of 25 square blocks. Thus, these
values form a data set representing the heat transfer map of each material.
Figure 6. The heat map of Aluminum, temperature ranging from 0K to 104K.
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2.3.3 Stage 3 Normalization and Comparison
After we achieve the data sets from both optical and thermal simulation, a
normalization process needs to be applied before synchronous comparison: First part of
normalization could be represented by this formula: Acolumn=Bcolumn-min(B), where B
column is the data set before this normalization process and A column is the normalized
data set. Since the sources of both thermal and optical models are at left top corner and
heat/light propagate from source corner to right bottom corner, the minimum value is
always the value of the right bottom corner. Then formula Ccolumn=Acolumn/(max(A)/100).
By these procedures above we can get each data set normalized from 0 to 100. Next, we
set the normalized data sets of three materials from thermal simulation as baseline,
comparing with Interconnect simulation data set by subtracting each other. So that the
average error can be presented after adding the errors together and dividing by 2500 (one
hundred times of the amount of data in each data set). The average accuracy is one minus
average error. Notably, limited by nowadays modulator technology, the maximum
extinction ratio of modulator can only be set to -10dB. Because the dissipating speed in
heat transfer is much faster than in optical simulation, the average accuracy is lower than
70% if the extinction ratio of all modulators is 0dB. Therefore, we can calculate the error
of each model then tune the modulator of these models and make their average accuracy
fixed to 70%.
To increase the calculating efficiency, I code an automatic tool with Python to
normalize data set and complete comparison between optical and thermal simulation. As
mentioned above in 2.3.1, the oscilloscope records 9998 values for every node, hence, in
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the script I appoint the tool to extract only the order of 9000 value for its stability. The
rest of the script are serving for normalization and subtracting one data set from the other
one. At the end, the tool would export a excel file recording the whole processing period.
The following figure is an example of one excel file.
Table 1. The excel file generated by python-scripted tool and converted to pdf file, which is the
accuracy of Calcarb cvd 20 comparing with one circuit. Column A is the No.9000 raw data extracted from oscilloscope output data set, which unit is W. Column B is the data from column A, converting unit to dB/m. Column C&D are the normalization process turns data set into 0-100. Column E is the 0-100 normalized data set of thermal simulation. After subtract column D by column E, we get the error distribution between two simulation. For calculating average error, we still need to take absolute number from column F preventing negative number. By adding 25 numbers in column G up and dividing them by 25, we got column H, which is the percentage of average error. Finally, subtracting 100 by the average error in column H, we can achieve the percentage of average accuracy we are figuring.
2.3.4 Result
Within the model, since we place S-parameter between every two nodes, it is
accessible for us to altering the ER of every modulator in terms of replacing the S-
parameter matrix calculated from power decay percentage. Initially, modulators near the
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laser source is raised to maximum value ----- 10dB, simply because the dissipating speed in heat map is much rapider than in optical model. According to the file exported by
comparison tool, we find out that more modulators need to be activated.
After 85 models built and adjusted, we finally match three PICs for three materials’
heat transfer map. As figure 8 shown below, the models with 70% average accuracy
matched with three materials’ heat map are structural different.
(a)
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(b)
(c)
Figure 7. (a) Model matched with Aluminum. (b) Model matched with Vanadium. (c) Model matched
with Calcarb.
Here we present a table on error map and average accuracy of different materials
matching with different models.
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Material Model (a) Error Map
and Accuracy
Model (b) Error Map
and Accuracy
Model (c) Error Map
and Accuracy
Aluminum
70.00%
67.26%
69.10%
Vanadium
70.90%
68.15%
70.00%
Calcarb
CVD 20
72.75%
70.00%
71.84%
Table 2. The error sets are input into Origin Lab software to produce the error maps. Calcarb CVD 20 has higher accuracy among three materials comparing with different circuits while Aluminum has lowest accuracy.
2.3.5. Analysis
In table 2, we can ascertain that the most considerable errors are on the first level
from the source. The main reason for that is because we set a restriction on the extinction
17
ratio of the modulator. Specifically, limited by current technology, the maximum
extinction ratio we choose in Interconnect simulation is -10dB. On the other hand, the
dissipate slope in thermal simulation is exceptionally steeper. Thus, even maximum
modulators are placed between source node and the first level nodes, the error after
analysis is remaining nonnegligible.
Besides, in the error map, we are not able to observe the error of source node
because our data set is normalized from 0 (right bottom node) to 100 (left top node).
Therefore, after subtraction, the error of these two nodes equals zero.
Moreover, due to our normalization process applied on optical simulation data set, if
the minimum value of data set is extremely small, which mean the power has almost fully
dissipated at the right bottom corner, the gaining values of first level nodes after addition
process would be exceedingly high, which causes greater error. In this case, a “free path”
needs to be left for light propagating through, otherwise light will die out before it
reached the right bottom corner. From another perspective, if high extinction ratio
modulators are added between boundary nodes, there should be no more modulator at
central “path”. Likewise, if central “path” was shut down, light should be able to
propagate through boundary.
2.4 The practical meaning of my work
The design we proposed delivers a possibility to replicate the functionality of a
lumped circuit model and solve PDE effortlessly and noniteratively by using finite
difference approach. Although the project is still on simulation stage1, but we restrict the
circuit with modern modulation technology, that we can experimentally solve various
PDE by nanofabrication. With reconfigurable part of ROC, we can now indicate that our
18
design is completely able to accommodate to different types of partial differential
equations.
19
Chapter 3 – Discussion 3.1 Limitation of ROC
Theoretically, electronic circuit are lumped elements while photonic circuit is
distributed network. Explicitly, in electronic circuit the wavelength of electromagnetic
wave is usually larger than the footprint size of the circuit. Thus, the phase difference
(delay) within the circuit is negligible that Kirchhoff’s law can be applied. Therefore, a
lumped circuit can solve problems with exactly 1:1 finite difference approach. On the
other hand, in photonic circuit, the wavelength of light involved is normally 1550nm. In
this way, the size of circuit is commonly bigger than the wavelength of light, hence
Kirchhoff’s law does not hold.
To sum up, in lumped circuit, local variation would cause global change within the
whole circuit, while in distributed circuit, local variation would only lead to local effect.
Hence, there is a fundamental gap in mapping electronic circuit and photonic circuit.
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